The linear independence of $1$, $\zeta (3)$, and $\zeta (4)$

In a recent breakthrough, Fauzan proved the irrationality of $\zeta (5)$. In this paper, we combine Fauzan’s determinant method with classical simultaneous approximations of zeta values studied by Zudilin and Dauguet–Zudilin to prove the linear independence of $1$, $\zeta (3)$, and $\zeta (4)$ over $\mathbb{Q}$.